Calculus and Differential Equations
Core concepts
Command differentiation rules, integration techniques, and the solution of first-order differential equations. Know separable equations, integrating factors for linear first-order equations, and how to apply initial conditions. Understand definite integrals as areas and the fundamental theorem of calculus.
Traps
- Forgetting the constant of integration or misapplying initial conditions.
- Separating variables without checking for constant solutions lost in division.
- Sign errors in integrating factors e^(integral of P dx).
- Confusing dy/dx notation with the chain rule when variables are implicit.
Worked example
Solve dy/dx=2xy with y(0)=3. Separate: dy/y=2x dx, giving ln|y|=x^2+C, so y=Ae^(x^2). Applying y(0)=3 gives A=3, hence y=3e^(x^2).
Rule of thumb
After solving any differential equation, substitute back and check both the equation and the initial condition.